What is the integral of sec(2x+3)tan(2x+3) ?

The integral of sec(2x+3)tan(2x+3) is 1/2 cos(3+2x)+1/2 sin(3+2x)tan(3+2x)+C

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integral of sec(2x+3)tan(2x+3)

\int sec (2 x + 3) tan (2 x + 3) d x

\frac{1}{2} cos (3 + 2 x) + \frac{1}{2} sin (3 + 2 x) tan (3 + 2 x) + C

Solution steps

\int sec (2 x + 3) tan (2 x + 3) d x

Rewrite using trig identities

= \int \frac{sin ( 2 x + 3 )}{cos ^{2} ( 2 x + 3 )} d x

Apply Integration By Parts

= - \int sin (3 + 2 x) d x + \frac{1}{2} sin (3 + 2 x) tan (3 + 2 x)

\int sin (3 + 2 x) d x = - \frac{1}{2} cos (3 + 2 x)

= - (- \frac{1}{2} cos (3 + 2 x)) + \frac{1}{2} sin (3 + 2 x) tan (3 + 2 x)

Simplify

= \frac{1}{2} cos (3 + 2 x) + \frac{1}{2} sin (3 + 2 x) tan (3 + 2 x)

Add a constant to the solution

= \frac{1}{2} cos (3 + 2 x) + \frac{1}{2} sin (3 + 2 x) tan (3 + 2 x) + C

What is the integral of sec(2x+3)tan(2x+3) ?
The integral of sec(2x+3)tan(2x+3) is 1/2 cos(3+2x)+1/2 sin(3+2x)tan(3+2x)+C